# International system of units

The International System of Units or SI ( French Système international d'unités ) is the most widely used system of units for physical quantities .

The SI defines physical units for selected quantities. The SI is based on seven basic units for corresponding basic quantities . The selection of the basic sizes and the definition of the associated basic units were based on practical aspects.

The SI is a metric system of units (i.e. a base unit is the meter ), it is decimal (i.e. the various units used to express a quantity differ only by whole powers of ten ) and it is a coherent system of units (i.e., each derived unit is a product of the powers of the base units without additional numerical factors).

The units of the international system of units are referred to as SI units in order to distinguish them from units in other systems of units. Since 2019, all SI units have been defined using natural constants .

## Dissemination and use

The SI is used all over the world. In most industrialized countries, its use for official and business transactions is required by law. An important exception is the USA, where the SI applies, but the Anglo-American measurement system (customary units) is also permitted in official and business transactions .

In addition to the SI units, other units are often used that are not SI units. The International Bureau of Weights and Measures (BIPM) itself defines a number of units that are "approved for use with the SI" , e.g. B. hectares , liters , minutes , hours and degrees . In addition, there are other legally approved units depending on the country , mostly for special purposes. In the European Union and Switzerland these are z. B. Tex and Diopter .

In some areas, units that differ from the SI are used: In shipping and aviation , non-SI-compliant units are used for altitude ( feet ), distances ( nautical miles ) and speeds ( knots ). In some areas of physics different natural units are used , in electrodynamics sometimes the Gaussian cgs system .

## Responsibilities

### International regulations

The International Bureau of Weights and Measures (Bureau International des Poids et Mesures, BIPM) and its General Conference on Weights and Measures (Conférence Générale des Poids et Mesures, CGPM) are responsible for international regulations on the SI . The brochure Le Système international d'unités published by the BIPM  - referred to as “the SI brochure” in German for short - is the reference set of rules . The 9th edition of the SI brochure was published in 2019.

### National implementation

The national metrological institutes are usually responsible for the national implementation of the SI . These are for example

An application of the SI arises only through laws or jurisdiction of individual states.

Laws regulating the introduction of the SI came into force in 1970 in the Federal Republic of Germany ( Units and Time Act ), 1973 in Austria ( Measure and Verification Act ), 1974 in the GDR and 1978 in Switzerland; In 1978 all transitional regulations regarding non-SI units were completed.

In the EU, the use of units in the field of legal metrology has been largely standardized through Directive 80/181 / EEC, among other things . In the European Union, Switzerland and most other countries, the use of the SI in official or business transactions is required by law. With Directive 2009/3 / EC, the use of additional units in the EU was permitted for an unlimited period (previous directives originally only allowed this until December 31, 2009). The main reason for this is not to hinder exports of goods to third countries.

## history

1790 : The French Academy of Sciences is commissioned by the French National Assembly to design a uniform system of weights and measures. It follows the principles of deriving the basic units from natural quantities, tracing all other units back to them and multiplying and subdividing all, with the exception of time, decimally. The following basic units are chosen:

1832 : Carl Friedrich Gauß develops a system of “absolute” electromagnetic units based on length (mm), mass (g) and time (s) with broken exponents.

1861 : A committee of the British Association for the Advancement of Science (BAAS) deals with the definition of electrical and magnetic units based on the work of Gauß and Weber , but with the basic units m (later cm), g, s. Because of the unwieldiness of the units obtained, “practical” units such as amperes, volts and ohms are introduced and it is decided that these must be exact decimal multiples of the basic units - e.g. B. 1 volt as 10 8 and 1 ohm as 10 9 electromagnetic cgs units . In the following decades these units established themselves worldwide. In 1894 the implementation of these units was standardized internationally.

1873 : James Clerk Maxwell suggests defining the units of length, time and mass using the wavelength and period of light and the mass of molecules.

1875 : The Meter Convention is signed by 17 states. The International Bureau for Weights and Measures is founded.

1889 : At the first General Conference on Weights and Measures (CGPM), the original measurements for the meter and the kilogram are recognized. The MKS system of units with the three base units meter (m), kilogram (kg) and second (s) is established.

1900 : Max Planck suggests defining basic units using physical "constants".

1901 : Giovanni Giorgi shows that the mechanical and electrical units can be merged into a coherent system with integer exponents by adding a fourth unit to the MBS system.

1935 : The International Electrotechnical Commission (IEC) adopts the Giorgi system, although the question of the fourth base unit remains unanswered.

1948 : The ampere is defined in the form valid until 2019.

1954 : The 10th CGPM adds three basic units to the MKS system: the ampere, the Kelvin (K) - until 1968 still referred to as degrees Kelvin (° K) - and the Candela (cd).

1960 : At the 11th CGPM, this extended MKS system was given the French name Système International d'Unités (SI) ("International System of Units").

1971 : At the 14th CGPM, the mole (mol) is added as the seventh and so far last base unit and is placed in the 6th position between Kelvin and Candela.

1979 : At the 16th CGPM, the candela receives its current definition and is linked to the watt . This connects the photometric units to the MKS system.

1983 : The 17th CGPM redefines the meter by assigning a fixed value to the speed of light .

2018 : The 26th CGPM resolves a fundamental reform with effect from May 20, 2019: All base units and therefore all units in general are now reduced to seven physical constants to which fixed values ​​are assigned. This makes the units independent of the implementation and its limited accuracy.

## SI units

There are seven basic units in the SI. All other SI units are derived from these base units.

### SI base units

The basic units of the SI and the corresponding basic sizes of the underlying size system ISQ were determined arbitrarily by the CGPM according to practical aspects. By definition, an SI base quantity cannot be expressed in terms of other base quantities. Similarly, an SI base unit cannot be expressed as the product of the powers of other base units.

In the international system of sizes and units, the seven basic sizes are expressed by the basic units second (s), meter (m), kilogram (kg), ampere (A), Kelvin (K), mol (mol) and candela (cd) and im SI defined in this order. Each base quantity is assigned a dimension with the same name. For example, the dimension of the base quantity length is also called length . The symbol of greatness is indicated by the letter " l " in italics ; that of the dimension with an upright, capitalized letter " L ". The practical realization of a dimension takes place through a corresponding coherent unit - in the case of the length through the meter.

Base size and
dimension name
Size
symbol
Dimension
symbol
unit unit-
sign
time t T second s
length l L. meter m
Dimensions m M. kilogram kg
Amperage I. I. amp A.
Thermodynamic
temperature
T Θ Kelvin K
Amount of substance n N Mole mol
Light intensity I v J Candela CD

### Derived quantities and units

All physical quantities except for the above seven basic quantities of the ISQ are derived quantities. Each physical quantity Q (. For English quantity ) has a dimension that can be displayed as a product of powers of the dimensions of the seven basic sizes:

dim QT α  ·  L β  ·  M γ  · I δ  ·  .theta .epsilon.  ·  N ζ  ·  J η

Each of the dimensional exponents α, β, γ, δ, ε, ζ and η is either zero or a positive or negative, generally integer . The magnitude of the exponent is usually between 0 and 4.

Correspondingly, the associated derived SI units can be expressed as the product of a numerical factor and the product of powers (power product) of the base units:

[ Q ] = 10 n  · s α  · m β  · kg γ  · A δ  · K ε  · mol ζ  · cd η

"[ Q ]" symbolically represents the expression "the unit of quantity  Q ", in accordance with the rules according to the VIM (International Vocabulary of Metrology - Basic and General Concepts and Associated Terms) published by the Joint Committee for Guides in Metrology .

### Coherent units

For practical reasons, the SI system offers several units for one size, which differ by an integer power of ten (the factor 10 n in the above formula). They are denoted by so-called SI prefixes such as kilo- (10 3 ) or milli- (10 −3 ). If the numerical factor is equal to one (i.e. when n  = 0), there is a coherent SI unit . Every physical quantity has exactly one coherent SI unit.

Examples:

• Meter (m) is the coherent SI unit of the base quantity “length”.
• Kilometers (km) is a non-coherent SI unit of the base quantity “length”.
• Meters per second (m / s) is the coherent SI unit of the derived quantity “speed”.
• Millimeter per second (mm / s) is a non-coherent SI unit of the derived quantity “speed”.
• Kilogram (kg) is the coherent SI unit of the base quantity "mass". It is the only base unit that has an SI prefix.
• Gram (g) is a non-coherent SI unit of the base quantity "mass".
• Light year is a unit of the base quantity "length", but not an SI unit.

An SI base unit is always the coherent unit of the associated base quantity. In addition, it can also serve as a coherent unit of derived quantities. Examples:

• The meter is the basic unit of the basic size "length". In addition, it can serve as a coherent derived unit for the amount of precipitation if it is expressed as volume per area in m 3 / m 2  = m.
• The ampere is the SI base unit of the electrical current strength and at the same time the coherent derived SI unit of the magnetic flux .

An advantage of the exclusive use of coherent SI units in physical and technical formulas is that no conversion factors are required between the units.

### SI derived units with a special name

22 coherent derived SI units were assigned their own names and unit symbols (symbols), which themselves can be combined with all base and derived units. For example, the SI unit of force, the Newton (= kg · m / s 2 ), is suitable for expressing the unit of energy, the Joule, as Newton times meter (N · m). The following table lists these 22 units in the same order as the SI brochure (9th edition).

size unit unit-
sign
in other
SI units
expressed
SI base
units off
down a)
flat angle Radians b) wheel m / m 1
Solid angle Steradian b) sr m 2 / m 2 1
frequency hertz Hz s −1
force Newton N J / m kg m s −2
pressure Pascal Pa N / m 2 kg m −1  s −2
Energy , work , amount of heat Joules J N · m; W · s kg m 2  s −2
power watt W. J / s; V · A kg m 2  s −3
electric charge Coulomb C. A · s
electrical voltage volt V W / A; J / C kg m 2  s −3  A −1
electrical capacitance farad F. C / V kg −1  m −2  s 4  A 2
electrical resistance ohm Ω V / A kg m 2  s −3  A −2
electrical conductance Siemens S. A / V kg −1  m −2  s 3  A 2
magnetic river Weber Wb V s kg m 2  s −2  A −1
magnetic flux density Tesla T Wb / m 2 kg s −2  A −1
Inductance Henry H Wb / A kg m 2  s −2  A −2
Celsius temperature Degrees Celsius c) ° C K
Luminous flux Lumens lm cd sr CD
Illuminance lux lx lm / m 2 cd m −2
Absorbed dose Gray Gy J / kg m 2  · s −2
Equivalent dose Sievert Sv J / kg m 2  · s −2
catalytic activity Catal cat mol s −1
a) Order according to the SI brochure
b)Radians (rad) and steradian (sr) can and are usually used instead of the unit 1 for the plane angle or the solid angle in order to emphasize the significance of the associated numerical value.
c)For the conversion of the Celsius temperature t into the thermodynamic temperature T, the following applies: t / ° C = T / K - 273.15. According to German unit law, degrees Celsius may not have any prefixes for units of measurement . The unit Kelvin is recommended for specifying temperature differences ; at Celsius temperatures, however, degrees Celsius may also be used.

### Definition of the SI units

#### Until 2018: Separately defined basic units

Up until 2018, each of the seven base units had its own definition: “The base unit X is the Y factor of ...” All other units were derived from this. These definitions have been changed several times with the advancing state of metrology and following revised fundamental considerations. For example, the meter was defined from 1889 on the basis of a prototype (" original meter ") and from 1960 onwards using a special light wavelength. With the definition , the implementation was specified at the same time , with some implementations depending on other basic units (e.g. the temperature was specified at which the length of the meter prototype was to be measured). When more suitable methods of implementation were developed, the definition of the corresponding base unit had to be changed in order to use them.

#### Since 2019: definition via physical constants

constant exact value since
Δ ν Cs Radiation from the cesium atom * 9 192 631 770 Hz 1967
c Speed ​​of Light 299 792 458 m / s 1983
H Planck's quantum of action 6th.626 070 15e-34 J · s 2019
e Elemental charge 1.602 176 634e-19th C. 2019
k B Boltzmann's constant 1.380 649e-23 J / K 2019
N A Avogadro's constant 6th.022 140 76e23 mol −1 2019
K cd Photometric radiation equivalent ** 683 lm / W 1979
* Hyperfine structure transition of the ground state of the cesium-133 atom **
for monochromatic radiation of the frequency 540 THz (green light)
Derivation of the SI base units from the precisely defined constants. The arrows mean "... is used to define ..." (but see note)

In November 2018, the 26th General Conference on Weights and Measures decided on a fundamental revision that came into force on May 20, 2019, World Metrology Day : After three of the basic units (s, m, cd) had previously been defined by the fact that one had assigned a fixed value to three physical constants (Δ ν Cs , c , K cd ), another four constants now got fixed values. Since then, no SI unit has been dependent on variable sizes or objects, and the implementation can be freely selected.

At the same time, the basic principle has been changed: since the reform, the seven basic definitions have been correspondingly: “The constant X has the numerical value Y if it is expressed in SI units.” All SI units can be derived from this; there is no longer any fundamental difference between base units and derived units. However, the term “base unit” continues to be used because it has proven useful to consistently use the same seven dimensions and their coherent units. The following table shows how these seven units can be derived from the seven defining constants:

unit Defining equation under the use of d. H. implicit from
second ${\ textstyle \ Delta \ nu _ {\ mathrm {Cs}} = 9 \, 192 \, 631 \, 770 \ {\ frac {1} {\ mathrm {s}}}}$
meter ${\ textstyle c = 299 \, 792 \, 458 \ {\ frac {\ mathrm {m}} {\ mathrm {s}}}}$ second Δ ν Cs
kilogram ${\ textstyle h = 6 {,} 626 \, 070 \, 15 \ cdot 10 ^ {- 34} \ {\ frac {\ mathrm {kg} \, \ mathrm {m} ^ {2}} {\ mathrm { s}}}}$ Second, meter Δ ν Cs , c
amp ${\ textstyle e = 1 {,} 602 \, 176 \, 634 \ cdot 10 ^ {- 19} \ \ mathrm {A} \, \ mathrm {s}}$ second Δ ν Cs
Kelvin ${\ textstyle k _ {\ mathrm {B}} = 1 {,} 380 \, 649 \ cdot 10 ^ {- 23} \ {\ frac {\ mathrm {kg} \, \ mathrm {m} ^ {2}} {\ mathrm {s} ^ {2} \, \ mathrm {K}}}}$ Second, meter, kilogram Δ ν Cs , h
Mole ${\ textstyle N _ {\ mathrm {A}} = 6 {,} 022 \, 140 \, 76 \ cdot 10 ^ {23} \ {\ frac {1} {\ mathrm {mol}}}}$
Candela ${\ textstyle K _ {\ mathrm {cd}} = 683 \ {\ frac {\ mathrm {cd} \, \ mathrm {sr} \, \ mathrm {s} ^ {3}} {\ mathrm {kg} \, \ mathrm {m} ^ {2}}}}$ Second, meter, kilogram Δ ν Cs , h

## Notation of sizes, numerical values ​​and units

The SI brochure and other normative documents not only specify the unit names, but also specify formatting rules for the spelling of unit symbols and numerical values.

### Consistent notation of quantities, units and numerical values

According to ISO, size symbols ( formula symbols ) must be written in italics , unit symbols in upright letters. Size information should always be given with a numerical value and unit; no multiplication sign in between:

A = { A } [ A ]

Here A stands as a symbol for the size, { A } for the numerical value of A and [ A ] for the unit of A (written out or as a unit symbol).

This coherent notation is deviated from if many similar size specifications are to be given in tables or axis labels. We recommend the notation A / [ A ] = { A }, e.g. B. T / K = 300, 400, 500 for T = 300 K, 400 K, 500 K. If the unit itself is a fraction, to avoid confusion it is recommended to put negative exponents individually after the unit symbols of the denominator, for a thermal resistance R in Kelvin per watt, ie “ R / K W −1 ”. The spellings " R in K / W", " R  (K / W)" and " R  [K / W]" are not standardized, but more common .

### Name and symbols of quantities

Size symbols ( formula symbols ) must be written in italics . The symbols can be freely chosen - common symbols such as l, m or t are only recommendations. DIN standards also contain recommendations for symbols. The SI brochure recommends choosing the name and symbol of a physical quantity without any association with a specific unit. Accordingly, terms such as liter output should be avoided. However, “ Celsius temperature ” and “ molar volume ” do not follow this recommendation. Further examples of non-compliance with this recommendation are the hour angle and the number of degree days .

Dimension symbols are written as upright capital letters in a sans serif font.

### Notation of units

Depending on the language, different spellings for unit names ( German second , English second , French seconde ) are possible. The unit names are also subject to the normal inflection of the respective language.

The unit symbols are internationally uniform. Regardless of the format of the surrounding text, they should be written in an upright font. They are written in lower case unless they are named after a person. For example, "1 s" means a sec, while "1 S" by the Werner von Siemens called Siemens represents. An exception to this rule is the non-SI unit liter : Although it is not named after a person, the capitalized “L” can be used for your unit symbol in addition to the lowercase “l”. The latter is particularly common in the Anglo-American region to avoid confusion with the number “one”.

An SI prefix (such as kilo- or milli- ) can be placed directly in front of the unit symbol of a coherent unit for a decimal multiple or part in order to represent units in different orders of magnitude more clearly. An exception is the kilogram (kg), which may only be used starting from the gram (g) with SI prefixes. For example, for 10 −6  kg it has to be “mg” and not “μkg”. For multiples of the kg, the use of the non-SI unit ton (1 t = 10 3  kg = 1 Mg) is permissible and customary, which in turn can be provided with prefixes such as kiloton (kt) or megaton (Mt).

Unit symbols follow the numerical value after a space; this also applies to percentages and temperatures in degrees Celsius. Only the unit symbols °, 'and " for the non-SI angle units degrees, minutes and seconds are placed directly after the numerical value without a gap.

References to certain facts should not be attached to unit symbols (as subscripts); however, they belong to the symbol of the physical quantity used or in explanatory text. Accordingly, V eff as “unit” of effective values of electrical voltage in volts or% (V / V) for “volume percent” is wrong .

In languages ​​that do not use the Latin writing system, it is sometimes common to write the unit symbols with characters from their own alphabet ( transcription ). For example, in Russian km is usually written as “км”, kg as “кг” and J as “Дж”.

### Notation of numerical values

The SI allows numbers to be divided into groups of three digits each, whereby the groups are not separated by periods or commas. Both the comma and the point are permitted as decimal separators ; Only the comma is standardized in German-speaking countries.

## literature

• E. Bodea: Giorgi's rational MBS system of measurements with dimensional coherence. 2nd Edition. Birkhäuser, 1949.
• The system of units. In: PTB-Mitteilungen 122 (2012) Heft 1, pp. 1-102. (online) (PDF) PDF, 5.8 MB

Commons : International system of units  - collection of images, videos and audio files

## Individual evidence

1. Text of the Unit Ordinance
2. DIN EN ISO 80000-3 : 2013 Sizes and units - Part 3: Space and time , Section 3-8.b
3. a b Le Système international d'unités . 9e édition, 2019 (the so-called "SI brochure", French and English).
4. The International System of Units (SI) . German translation of the BIPM brochure "Le Système international d'unités / The International System of Units (8e édition, 2006)". In: PTB-Mitteilungen . tape  117 , no. 2 , 2007 ( Online [PDF; 1.4 MB ]). - Note: This is the translation of the 2006 SI brochure; the translation of the current version is not yet available.
5. Directive 2009/3 / EC of the European Parliament and of the Council of March 11, 2009 amending Directive 80/181 / EEC of the Council on the harmonization of the laws of the member states on units in metrology
6. ^ JC Maxwell: A Treatise on Electricity and Magnetism . Clarendon Press, Oxford 1873, Vol. 1 pp. 3-4; Wikisource
7. "On the other hand, it should not be without interest to note that with the help of the two [...] constants a and b, it is possible to set up units for length, mass, time and temperature, which, independent of special bodies and substances, keep their meaning for all times and for all, also extraterrestrial and extra-human cultures necessary and which can therefore be described as 'natural units of measurement'. "- M. Planck. In: Ann. Physik , 1, 1900, p. 69; according to: natural constants as the main actor
8. ^ J. de Boer: Giorgi and the International System of Units. In: C. Egidi (Ed.): Giovanni Giorgi and his contribution to electrical metrology. Politecnico, Torino 1990, pp. 33-39.
9. a b c Resolution 1 of the 26th CGPM (2018). In: bipm.org. Bureau International des Poids et Mesures , accessed February 24, 2020 .
10. 26th CGPM (2018) - Resolutions adopted / Résolutions adoptées. (PDF; 1.2 MB) Versailles 13–16 November 2018. In: bipm.org. Bureau International des Poids et Mesures, November 19, 2018, pp. 2–5 , accessed on April 28, 2019 (English, French).
11. The new International System of Units (SI) (PDF; 4.7 MB) PTB information sheet with an explanation and description of the new definition of the base units 2019, accessed on June 22, 2019
12. “Prior to the definitions adopted in 2018, the SI was defined through seven base units from which the derived units were constructed as products of powers of the base units. Defining the SI by fixing the numerical values ​​of seven defining constants has the effect that this distinction is, in principle, not needed [...] Nevertheless, the concept of base and derived units is maintained because it is useful and historically well established [...] ” , SI brochure, Chapter 2.3 bipm.org (PDF)
13. New definitions in the International System of Units (SI). (PDF; 1.3 MB) PTB , accessed on October 31, 2019 .
14. ^ Langenscheidt: Dictionary German-Russian. Retrieved July 2, 2019 . , see also table in Russian Wikipedia
15. ^ Resolution 10 of the 22nd CGPM (2003). In: bipm.org. Bureau International des Poids et Mesures , accessed February 24, 2020 .
16. DIN EN ISO 80000-1: 2013-08, sizes and units - Part 1: General; German version of EN ISO 80000-1: 2013.

## Remarks

1. It is sometimes said that the SI does not apply in the USA. This is not the case: since the Metric Act of 1866 , expanded to include the SI in 2007, the metric system has been approved in the USA. It has been the preferred measurement system for US trade and commerce ( Metric Conversion Act law.cornell.edu ) since 1975 , but it is not mandatory. For trade with end consumers, a federal law (Fair Packaging and Labeling Act) prescribes labeling in both metric and customary units .
2. a b In c , h , e and k B If it is fundamental physical constants . Δν Cs is a universally reproducible frequency that is independent of any implementation rule . N A is a numerical value determined by agreement, which should correspond as precisely as possible to the conversion factor between the atomic mass unit and the unit “gram”. K cd is also an arbitrarily determined conversion factor between physical and photobiological quantities ( publications.csiro.au ).
3. a b c The units "Kelvin" and "Candela" do not depend on the speed of light c . Although its definition depends on the illustration shown here u. a. from the units “meter” and “kilogram” and these in turn from c . However, if the Kelvin and Candela are entirely based on the defining constants of the SI, then c is eliminated in the calculation .
4. There are derived units that are “more direct”, i.e. H. are defined by fewer constants than basic units: The unit “coulomb” is defined solely by the constant e , for the “ampere” you also need Δ ν Cs . For “Joule” and “Watt” only h and Δ ν Cs are required, for the “Kilogram” additionally c .